Differential Equations : Linear, Nonlinear, Ordinary, Partial

Finding And Interpreting The Solutions Of Differential Equations Is A Central And Essential Part Of Applied Mathematics. This Book Aims To Enable The Reader To Develop The Required Skills Needed For A Thorough Understanding Of The Subject. The Authors Focus On The Business Of Constructing Solutions Analytically, And Interpreting Their Meaning, Using Rigorous Analysis Where Needed. Matlab Is Used Extensively To Illustrate The Material. There Are Many Worked Examples Based On Interesting And Unusual Real World Problems. A Large Selection Of Exercises Is Provided, Including Several Lengthier Projects, Some Of Which Involve The Use Of Matlab. The Coverage Is Broad, Ranging From Basic Second-order Odes And Pdes, Through To Techniques For Nonlinear Differential Equations, Chaos, Asymptotics And Control Theory. This Broad Coverage, The Authors' Clear Presentation And The Fact That The Book Has Been Thoroughly Class-tested Will Increase Its Attraction To Undergraduates At Each Stage Of Their Studies. A.c. King, J. Billingham And S.r. Otto. Includes Bibliographical References And Index. Cover Half-title Title Copyright Contents Preface Part One Linear Equations CHAPTER ONE Variable Coefficient, Second Order, Linear, Ordinary Differential Equations 1.1 The Method of Reduction of Order 1.2 The Method of Variation of Parameters 1.2.1 The Wronskian 1.3 Solution by Power Series: The Method of Frobenius 1.3.1 The Roots of the Indicial Equation Differ by an Integer 1.3.2 The Roots of the Indicial Equation Differ by a Noninteger Quantity 1.3.3 The Roots of the Indicial Equation are Equal 1.3.4 Singular Points of Differential Equations 1.3.5 An outline proof of Theorem 1.3 1.3.6 The point at infinity CHAPTER TWO Legendre Functions 2.1 Definition of the Legendre Polynomials, p(x) 2.2 The Generating Function for P(x) 2.3 Differential and Recurrence Relations Between Legendre Polynomials 2.4 Rodrigues’ Formula 2.5 Orthogonality of the Legendre Polynomials 2.6 Physical Applications of the Legendre Polynomials 2.6.1 Heat Conduction 2.6.2 Fluid Flow 2.7 The Associated Legendre Equation CHAPTER THREE Bessel Functions 3.1 The Gamma Function and the Pockhammer Symbol 3.2 Series Solutions of Bessel’s Equation 3.3 The Generating Function for J(x), n an integer 3.4 Differential and Recurrence Relations Between Bessel Functions 3.5 Modifled Bessel Functions 3.6 Orthogonality of the Bessel Functions 3.7 Inhomogeneous Terms in Bessel’s Equation 3.8 Solutions Expressible as Bessel Functions 3.9 Physical Applications of the Bessel Functions 3.9.1 Vibrations of an Elastic Membrane 3.9.2 Frequency Modulation (FM) CHAPTER FOUR Boundary Value Problems, Green’s Functions and Sturm–Liouville Theory 4.1 Inhomogeneous Linear Boundary Value Problems 4.1.1 Solubility 4.1.2 The Green’s Function 4.2 The Solution of Boundary Value Problems by Eigenfunction Expansions 4.2.1 Self-Adjoint Operators 4.2.2 Boundary Conditions 4.2.3 Eigenvalues and Eigenfunctions of Hermitian Linear 4.2.4 Eigenfunction Expansions 4.3 Sturm–Liouville Systems 4.3.1 The Sturm–Liouville Equation 4.3.2 Boundary Conditions 4.3.3 Properties of the Eigenvalues and Eigenfunctions 4.3.4 Bessel’s Inequality, Approximation in the Mean and Completeness 4.3.5 Further Properties of Sturm–Liouville Systems 4.3.6 Two Examples from Quantum Mechanics CHAPTER FIVE Fourier Series and the Fourier Transform 5.1 General Fourier Series 5.2 The Fourier Transform 5.2.1 Generalized Functions 5.2.2 Derivatives of Generalized Functions 5.2.3 Fourier Transforms of Generalized Functions 5.2.4 The Inverse Fourier Transform 5.2.5 Transforms of Derivatives and Convolutions 5.3 Green’s Functions Revisited 5.4 Solution of Laplace’s Equation Using Fourier Transforms 5.5 Generalization to Higher Dimensions 5.5.1 The Delta Function in Higher Dimensions 5.5.2 Fourier Transforms in Higher Dimensions CHAPTER SIX Laplace Transforms 6.1 Definition and Examples 6.1.1 The Existence of Laplace Transforms 6.2 Properties of the Laplace Transform 6.3 The Solution of Ordinary Differential Equations Using Laplace Transforms 6.3.1 The Convolution Theorem 6.4 The Inversion Formula for Laplace Transforms CHAPTER SEVEN Classification, Properties and Complex Variable Methods for Second Order Partial Differential Equations 7.1 Classification and Properties of Linear, Second Order Partial Differential Equations in Two Independent Variables 7.1.1 Classification 7.1.2 Canonical Forms 7.1.3 Properties of Hyperbolic Equations 7.1.4 Properties of Elliptic Equations 7.1.5 Properties of Parabolic Equations 7.2 Complex Variable Methods for Solving Laplace’s Equation 7.2.1 The Complex Potential 7.2.2 Simple Flows Around Blunt Bodies 7.2.3 Conformal Transformations Part Two Nonlinear Equations and Advanced Techniques CHAPTER EIGHT Existence, Uniqueness, Continuity and Comparison of Solutions of Ordinary Differential Equations 8.1 Local Existence of Solutions 8.2 Uniqueness of Solutions 8.3 Dependence of the Solution on the Initial Conditions 8.4 Comparison Theorems CHAPTER NINE Nonlinear Ordinary Differential Equations: Phase Plane Methods 9.1 Introduction: The Simple Pendulum 9.2 First Order Autonomous Nonlinear Ordinary Differential Equations 9.2.1 The Phase Line 9.2.2 Local Analysis at an Equilibrium Point 9.3 Second Order Autonomous Nonlinear Ordinary Differential Equations 9.3.1 The Phase Plane 9.3.2 Equilibrium Points 9.3.3 An Example from Mechanics 9.3.4 Example: Population Dynamics 9.3.5 The Poincaré Index 9.3.6 Bendixson’s Negative Criterion and Dulac’s Extension 9.3.7 The Poincaré–Bendixson Theorem 9.3.8 The Phase Portrait at Infinity 9.3.9 A Final Example: Hamiltonian Systems 9.4 Third Order Autonomous Nonlinear Ordinary Differential Equations CHAPTER TEN Group Theoretical Methods 10.1 Lie Groups 10.1.1 The Infinitesimal Transformation 10.1.2 Infinitesimal Generators and the Lie Series 10.2 Invariants Under Group Action 10.3 The Extended Group 10.4 Integration of a First Order Equation with a Known Group Invariant 10.5 Towards the Systematic Determination of Groups Under Which a First Order Equation is Invariant 10.6 Invariants for Second Order Differential Equations 10.7 Partial Differential Equations CHAPTER ELEVEN Asymptotic Methods: Basic Ideas 11.1 Asymptotic Expansions 11.1.1 Gauge Functions 11.1.2 Example: Series Expansions of the Exponential Integral, Ei(x) 11.1.3 Asymptotic Sequences of Gauge Functions 11.2 The Asymptotic Evaluation of Integrals 11.2.1 Laplace’s Method 11.2.2 The Method of Stationary Phase 11.2.3 The Method of Steepest Descents CHAPTER TWELVE Asymptotic Methods: Differential Equations 12.1 An Instructive Analogy: Algebraic Equations 12.1.1 Example: A Regular Perturbation 12.1.2 Example: A Singular Perturbation 12.2 Ordinary Differential Equations 12.2.1 Regular Perturbations 12.2.2 The Method of Matched Asymptotic Expansions Van Dyke’s Matching Principle Composite Expansions Interior Layers 12.2.3 Nonlinear Problems 12.2.4 The Method of Multiple Scales 12.2.5 Slowly Damped Nonlinear Oscillations: Kuzmak’s Method 12.2.6 The Effect of Fine Scale Structure on Reaction–Diffusion Processes 12.2.7 The WKB Approximation 12.3 Partial Differential Equations CHAPTER THIRTEEN Stability, Instability and Bifurcations 13.1 Zero Eigenvalues and the Centre Manifold Theorem 13.1.1 Construction of the Centre Manifold 13.1.2 The Stable, Unstable and Centre Manifolds 13.2 Lyapunov’s Theorems 13.3 Bifurcation Theory 13.3.1 First Order Ordinary Differential Equations 13.3.2 Second Order Ordinary Differential Equations 13.3.3 Global Bifurcations CHAPTER FOURTEEN Time-Optimal Control in the Phase Plane 14.1 Definitions 14.2 First Order Equations 14.3 Second Order Equations 14.3.1 Properties of sets of points in the plane 14.3.2 Matrix solution of systems of constant coefficient ordinary differential equations 14.4 Examples of Second Order Control Problems 14.5 Properties of the Controllable Set 14.6 The Controllability Matrix 14.7 The Time-Optimal Maximum Principle (TOMP) CHAPTER FIFTEEN An Introduction to Chaotic Systems 15.1 Three Simple Chaotic Systems 15.1.1 A Mechanical Oscillator 15.1.2 A Chemical Oscillator 15.1.3 The Lorenz Equations 15.2 Mappings 15.2.1 Fixed and Periodic Points of Maps 15.2.2 Tents and Horseshoes 15.3 The Poincaré Return Map 15.4 Homoclinic Tangles 15.4.1 Mel’nikov Theory 15.4.2 Unperturbed System (Epsilon = 0) 15.4.3 Perturbed System... 15.5 Quantifying Chaos: Lyapunov Exponents and the Lyapunov Spectrum 15.5.1 Lyapunov Exponents of Systems of Ordinary Differential Equations 15.5.2 The Lyapunov Spectrum APPENDIX 1 Linear Algebra A1.1 Vector Spaces Over the Real Numbers A1.2 Inner Product Spaces A1.3 Linear Transformations and Matrices A1.4 The Eigenvalues and Eigenvectors of a Matrix APPENDIX 2 Continuity and Differentiability APPENDIX 3 Power Series A3.1 Maclaurin Series A3.2 Taylor Series A3.3 Convergence of Power Series A3.4 Taylor Series for Functions of Two Variables APPENDIX 4 Sequences of Functions APPENDIX 5 Ordinary Differential Equations A5.1 Variables Separable A5.2 Integrating Factors A5.3 Second Order Equations with Constant Coefficients APPENDIX 6 Complex Variables A6.1 Analyticity and the Cauchy–Riemann Equations A6.2 Cauchy’s Theorem, Cauchy’s Integral Formula and Taylor’s Theorem A6.3 The Laurent Series and Residue Calculus A6.4 Jordan’s Lemma A6.5 Linear Ordinary Differential Equations in the Complex Plane APPENDIX 7 A Short Introduction to MATLAB A7.1 Getting Started A7.2 Variables, Vectors and Matrices A7.3 User-Defined Functions A7.4 Graphics A7.5 Programming in MATLAB Bibliography Index Part one: Linear equations 1. Variable coefficient, second order, linear, ordinary differential equations 2. Legendre functions 3. Bessel functions 4. Boundary value problems, Green's functions and Sturm-Liouville theory 5. Fourier series and the fourier transform 6. Laplace transforms 7. Classification, properties and complex variable methods for second order partial differential equations Part two: Nonlinear equations and advanced techniques 8. Existence, uniqueness, continuity and comparison of solutions of ordinary differential equations 9. Nonlinear ordinary differential equations: Phase plane methods 10. Group theoretical methods 11. Asymptotic methods: Basic ideas 12. Asymptotic methods: Differential equations 13. Stability, instability and bifurcations 14. Time-optimal control in the phase plane 15. Introduction to chaotic systems. Differential equations are vital to science, engineering and mathematics, and this book enables the reader to develop the required skills needed to understand them thoroughly. The authors focus on constructing solutions analytically and interpreting their meaning and use MATLAB extensively to illustrate the material along with many examples based on interesting and unusual real world problems. A large selection of exercises is also provided. The authors focus on constructing solutions analytically, and interpreting their meaning; MATLAB is used extensively to illustrate the material. The many worked examples, based on interesting real world problems, the large selection of exercises, including several lengthier projects, the broad coverage, and clear and concise presentation will appeal to undergraduates.